Peterson conjecture via Lagrangian correspondences and wonderful compactifications

TL;DR

This paper investigates the Floer cohomology of cotangent fibers and diagonals in certain symplectic manifolds related to Lie groups, establishing an isomorphism after localization through pseudo-holomorphic quilt counts.

Contribution

It computes the leading term of an $A_{ abla}$-homomorphism between Floer cohomologies using pseudo-holomorphic quilts and proves their ring isomorphism after localization.

Findings

Computed the leading term of the $A_{ abla}$-homomorphism.

Established isomorphism of Floer cohomologies as rings after localization.

Used pseudo-holomorphic quilt counting techniques.

Abstract

For a simply-connected compact semisimple Lie group $G$ and its maximal torus $T$, we study the $A_{\infty}$-functor associated to the moment Lagrangian correspondence from the cotangent bundle $T^*G$ to the square $G/T^{-} \times G/T$. In particular, we compute the leading term of the $A_{\infty}$-homomorphism from the wrapped Floer cohomology $HW^*(T^*_e G, T^*_e G)$ of the cotangent fiber $T_e^*G$ to the Floer cohomology $HF^*(\Delta, \Delta)$ of the diagonal $\Delta$ in the square $G/T^{-} \times G/T$ by determining the count of certain pseudo-holomorphic quilts. As a consequence, we prove that the Floer cohomologies $HW^*(T^*_e G, T^*_e G)$ and $HF^*(\Delta,\Delta)$ are isomorphic as rings after a localization.